In the mathematical theory of
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their ...
s, a quasi-Fuchsian group is a Kleinian group whose
limit set
In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they ca ...
is contained in an invariant
Jordan curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, and otherwise it is said to be of type two. Some authors use "quasi-Fuchsian group" to mean "quasi-Fuchsian group of type 1", in other words the limit set is the whole Jordan curve. This terminology is incompatible with the use of the terms "type 1" and "type 2" for Kleinian groups: all quasi-Fuchsian groups are Kleinian groups of type 2 (even if they are quasi-Fuchsian groups of type 1), as their limit sets are proper subsets of the Riemann sphere. The special case when the Jordan curve is a circle or line is called a
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
, named after
Lazarus Fuchs
Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a Jewish-German mathematician who contributed important research in the field of linear differential equations. He was born in Moschin (Mosina) (located in Grand Duchy of Posen) and ...
by Henri Poincaré.
Finitely generated quasi-Fuchsian groups are conjugate to Fuchsian groups under quasi-conformal transformations.
The space of quasi-Fuchsian groups of the first kind is described by the
simultaneous uniformization theorem of Bers.
References
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External links
A picture of the limit set of a quasi-Fuchsian groupfrom {{harv, Fricke, Klein, 1897, p= 418.
A picture of the limit set of a quasi-Fuchsian groupA picture of the limit set of a quasi-Fuchsian group
Kleinian groups